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Calculus Examples
Step 1
Differentiate both sides of the equation.
Step 2
The derivative of with respect to is .
Step 3
Step 3.1
Differentiate using the chain rule, which states that is where and .
Step 3.1.1
To apply the Chain Rule, set as .
Step 3.1.2
The derivative of with respect to is .
Step 3.1.3
Replace all occurrences of with .
Step 3.2
Differentiate using the chain rule, which states that is where and .
Step 3.2.1
To apply the Chain Rule, set as .
Step 3.2.2
Differentiate using the Power Rule which states that is where .
Step 3.2.3
Replace all occurrences of with .
Step 3.3
Differentiate using the Constant Multiple Rule.
Step 3.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.3.2
Multiply by .
Step 3.4
Differentiate using the Quotient Rule which states that is where and .
Step 3.5
Differentiate.
Step 3.5.1
Differentiate using the Power Rule which states that is where .
Step 3.5.2
Move to the left of .
Step 3.5.3
By the Sum Rule, the derivative of with respect to is .
Step 3.5.4
Since is constant with respect to , the derivative of with respect to is .
Step 3.5.5
Differentiate using the Power Rule which states that is where .
Step 3.5.6
Multiply by .
Step 3.5.7
Since is constant with respect to , the derivative of with respect to is .
Step 3.5.8
Simplify the expression.
Step 3.5.8.1
Add and .
Step 3.5.8.2
Multiply by .
Step 3.6
Multiply by by adding the exponents.
Step 3.6.1
Move .
Step 3.6.2
Use the power rule to combine exponents.
Step 3.6.3
Add and .
Step 3.7
Combine and .
Step 3.8
Move to the left of .
Step 3.9
Simplify.
Step 3.9.1
Apply the product rule to .
Step 3.9.2
Apply the product rule to .
Step 3.9.3
Apply the product rule to .
Step 3.9.4
Apply the product rule to .
Step 3.9.5
Apply the distributive property.
Step 3.9.6
Apply the distributive property.
Step 3.9.7
Apply the distributive property.
Step 3.9.8
Combine terms.
Step 3.9.8.1
Raise to the power of .
Step 3.9.8.2
Multiply the exponents in .
Step 3.9.8.2.1
Apply the power rule and multiply exponents, .
Step 3.9.8.2.2
Multiply by .
Step 3.9.8.3
Multiply by the reciprocal of the fraction to divide by .
Step 3.9.8.4
Multiply by .
Step 3.9.8.5
Multiply by .
Step 3.9.8.6
Raise to the power of .
Step 3.9.8.7
Use the power rule to combine exponents.
Step 3.9.8.8
Add and .
Step 3.9.8.9
Multiply by .
Step 3.9.8.10
Multiply by .
Step 3.9.8.11
Multiply by .
Step 3.9.8.12
Multiply by .
Step 3.9.8.13
Subtract from .
Step 3.9.8.14
Raise to the power of .
Step 3.9.8.15
Multiply the exponents in .
Step 3.9.8.15.1
Apply the power rule and multiply exponents, .
Step 3.9.8.15.2
Multiply by .
Step 3.9.8.16
Multiply by .
Step 3.9.8.17
Multiply by by adding the exponents.
Step 3.9.8.17.1
Use the power rule to combine exponents.
Step 3.9.8.17.2
Add and .
Step 3.9.8.18
Move to the left of .
Step 3.9.8.19
Multiply by .
Step 3.9.8.20
Move to the left of .
Step 3.9.8.21
Cancel the common factor of and .
Step 3.9.8.21.1
Factor out of .
Step 3.9.8.21.2
Cancel the common factors.
Step 3.9.8.21.2.1
Factor out of .
Step 3.9.8.21.2.2
Cancel the common factor.
Step 3.9.8.21.2.3
Rewrite the expression.
Step 3.9.8.22
Cancel the common factor of and .
Step 3.9.8.22.1
Factor out of .
Step 3.9.8.22.2
Cancel the common factors.
Step 3.9.8.22.2.1
Factor out of .
Step 3.9.8.22.2.2
Cancel the common factor.
Step 3.9.8.22.2.3
Rewrite the expression.
Step 3.9.8.23
Cancel the common factor of and .
Step 3.9.8.23.1
Factor out of .
Step 3.9.8.23.2
Cancel the common factors.
Step 3.9.8.23.2.1
Factor out of .
Step 3.9.8.23.2.2
Cancel the common factor.
Step 3.9.8.23.2.3
Rewrite the expression.
Step 3.9.8.24
Cancel the common factor of and .
Step 3.9.8.24.1
Factor out of .
Step 3.9.8.24.2
Cancel the common factors.
Step 3.9.8.24.2.1
Factor out of .
Step 3.9.8.24.2.2
Cancel the common factor.
Step 3.9.8.24.2.3
Rewrite the expression.
Step 3.9.8.25
Cancel the common factor of and .
Step 3.9.8.25.1
Factor out of .
Step 3.9.8.25.2
Cancel the common factors.
Step 3.9.8.25.2.1
Cancel the common factor.
Step 3.9.8.25.2.2
Rewrite the expression.
Step 3.9.9
Reorder terms.
Step 3.9.10
Simplify the numerator.
Step 3.9.10.1
Factor out of .
Step 3.9.10.1.1
Factor out of .
Step 3.9.10.1.2
Factor out of .
Step 3.9.10.1.3
Factor out of .
Step 3.9.10.2
Rewrite the expression using the negative exponent rule .
Step 3.9.10.3
Combine and .
Step 3.9.10.4
To write as a fraction with a common denominator, multiply by .
Step 3.9.10.5
Combine and .
Step 3.9.10.6
Combine the numerators over the common denominator.
Step 3.9.10.7
Combine exponents.
Step 3.9.10.7.1
Combine and .
Step 3.9.10.7.2
Combine and .
Step 3.9.10.8
Remove unnecessary parentheses.
Step 3.9.10.9
Reduce the expression by cancelling the common factors.
Step 3.9.10.9.1
Factor out of .
Step 3.9.10.9.2
Factor out of .
Step 3.9.10.9.3
Cancel the common factor.
Step 3.9.10.9.4
Rewrite the expression.
Step 3.9.11
Multiply the numerator by the reciprocal of the denominator.
Step 3.9.12
Multiply by .
Step 3.9.13
Factor out of .
Step 3.9.14
Rewrite as .
Step 3.9.15
Factor out of .
Step 3.9.16
Rewrite as .
Step 3.9.17
Move the negative in front of the fraction.
Step 4
Reform the equation by setting the left side equal to the right side.
Step 5
Replace with .